Extending the Oracle Approximating Shrinkage to General Distributions
Abstract
The Oracle Approximating Shrinkage (OAS) estimator provides a closed-form, data-driven shrinkage intensity for regularizing sample covariance matrices toward a scaled identity target, but its derivation assumes Gaussian data. In practice, financial returns, neural signals, and many other high-dimensional datasets exhibit heavy tails and non-zero kurtosis, violating this assumption. We extend OAS to any distribution with finite fourth moments by incorporating a correction term α built from the fourth-order joint cumulants of the data. The correction vanishes under Gaussianity, recovering the OAS formula as a special case; it admits a consistent O(np) sample estimator and satisfies ρ∈ [0, 1] exactly for both the population cumulant and its sample analogue. Simulations and an equity-returns study confirm that the correction closes the gap between OAS and the Ledoit-Wolf estimator under non-Gaussian data, with OAS-Extended performing comparably across all tested settings. The contribution is a principled correction for OAS practitioners facing heavy-tailed data; structurally, it is distinct from prior work extending Ledoit-Wolf to elliptical distributions, which optimizes against a fixed oracle target and yields a different closed-form formula.
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BibTeX
@misc{lewis2026extending,
title = {{Extending the Oracle Approximating Shrinkage to General Distributions}},
author = {Michael J. Lewis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22144},
eprint = {2610.22144},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22144},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Extending the Oracle Approximating Shrinkage to General Distributions
AU - Michael J. Lewis
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22144
UR - https://arxiv.org/abs/2610.22144
AB - The Oracle Approximating Shrinkage (OAS) estimator provides a closed-form, data-driven shrinkage intensity for regularizing sample covariance matrices toward a scaled identity target, but its derivation assumes Gaussian data. In practice, financial returns, neural signals, and many other high-dimensional datasets exhibit heavy tails and non-zero kurtosis, violating this assumption. We extend OAS to any distribution with finite fourth moments by incorporating a correction term α built from the fourth-order joint cumulants of the data. The correction vanishes under Gaussianity, recovering the OAS formula as a special case; it admits a consistent O(np) sample estimator and satisfies ρ∈ [0, 1] exactly for both the population cumulant and its sample analogue. Simulations and an equity-returns study confirm that the correction closes the gap between OAS and the Ledoit-Wolf estimator under non-Gaussian data, with OAS-Extended performing comparably across all tested settings. The contribution is a principled correction for OAS practitioners facing heavy-tailed data; structurally, it is distinct from prior work extending Ledoit-Wolf to elliptical distributions, which optimizes against a fixed oracle target and yields a different closed-form formula.
ER -